Pseudovarieties generated by Brauer type monoids K. Auinger NSAC - - PowerPoint PPT Presentation

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Pseudovarieties generated by Brauer type monoids K. Auinger NSAC - - PowerPoint PPT Presentation

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product Pseudovarieties generated by Brauer type monoids K. Auinger NSAC 2013 K. Auinger Pseudovarieties generated by

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Pseudovarieties generated by Brauer type monoids

  • K. Auinger

NSAC 2013

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

all monoids (semigroups) in this talk are finite

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

all monoids (semigroups) in this talk are finite Bn = all partitions of {1, . . . , n, 1′, . . . , n′} into blocks of size 2

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

all monoids (semigroups) in this talk are finite Bn = all partitions of {1, . . . , n, 1′, . . . , n′} into blocks of size 2 members of Bn are diagrams like this: 7 7′ 1 1′

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams =

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams =

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 10

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams = =

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 11

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Composition of diagrams = = composition of diagrams defines a monoid structure on Bn, the Brauer monoid

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Jn = all elements of Bn whose diagrams can be drawn without intersecting lines:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Jn = all elements of Bn whose diagrams can be drawn without intersecting lines:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Jn = all elements of Bn whose diagrams can be drawn without intersecting lines: Jn is closed under composition of diagrams, is called the Jones monoid or the Temperley–Lieb monoid

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}?

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}?

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}?

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly,

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m Bn × Bm ֒ → Bn+m.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m Bn × Bm ֒ → Bn+m.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m Bn × Bm ֒ → Bn+m. Hence,

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m Bn × Bm ֒ → Bn+m. Hence, pvar{Jn | n ∈ N} = the divisors of all Jn

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Question What is pvar{Jn | n ∈ N}? What is pvar{Bn | n ∈ N}? Clearly, Jn × Jm ֒ → Jn+m Bn × Bm ֒ → Bn+m. Hence, pvar{Jn | n ∈ N} = the divisors of all Jn pvar{Bn | n ∈ N} = the divisors of all Bn.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A pvar{Bn | n ∈ N} = M.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A pvar{Bn | n ∈ N} = M.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A pvar{Bn | n ∈ N} = M. In other words,

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A pvar{Bn | n ∈ N} = M. In other words, each aperiodic monoid divides some Jn,

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

A = all aperiodic monoids M = all monoids. Theorem pvar{Jn | n ∈ N} = A pvar{Bn | n ∈ N} = M. In other words, each aperiodic monoid divides some Jn, each monoid divides some Bn.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n Problem J.-´

  • E. Pin (1987): Is it true that pvar{On | n ∈ N} = A?
  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n Problem J.-´

  • E. Pin (1987): Is it true that pvar{On | n ∈ N} = A?

Higgins (1995): No, there exist aperiodic monoids that do not divide any On.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n Problem J.-´

  • E. Pin (1987): Is it true that pvar{On | n ∈ N} = A?

Higgins (1995): No, there exist aperiodic monoids that do not divide any On. Almeida, Volkov (1998): {On | n ∈ N} is far away from a generating series of A

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n Problem J.-´

  • E. Pin (1987): Is it true that pvar{On | n ∈ N} = A?

Higgins (1995): No, there exist aperiodic monoids that do not divide any On. Almeida, Volkov (1998): {On | n ∈ N} is far away from a generating series of A

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition On = all order preserving mappings of the chain 1 < 2 < · · · < n Problem J.-´

  • E. Pin (1987): Is it true that pvar{On | n ∈ N} = A?

Higgins (1995): No, there exist aperiodic monoids that do not divide any On. Almeida, Volkov (1998): {On | n ∈ N} is far away from a generating series of A: the interval [O, A] where O := pvar{On | n ∈ N} is very big.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

  • J2n = On, the involutory monoid generated by On (w.r.t.

reflection of diagrams along the vertical axis as involution)

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

  • J2n = On, the involutory monoid generated by On (w.r.t.

reflection of diagrams along the vertical axis as involution) {On | n ∈ N} is not so far from a generating series of A since {On | n ∈ N} is one

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

On can be viewed as a submonoid of J2n:

  • J2n = On, the involutory monoid generated by On (w.r.t.

reflection of diagrams along the vertical axis as involution) {On | n ∈ N} is not so far from a generating series of A since {On | n ∈ N} is one: Jean-´ Eric pointed in the right direction!

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})
  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})
  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})

The proof shows that pvar{Jn | n ∈ N} is closed under the

  • peration

S → S ≀ ([2], U2),

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})

The proof shows that pvar{Jn | n ∈ N} is closed under the

  • peration

S → S ≀ ([2], U2),

  • r, more precisely:
  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 53

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})

The proof shows that pvar{Jn | n ∈ N} is closed under the

  • peration

S → S ≀ ([2], U2),

  • r, more precisely:

S ≺ Jn = ⇒ S ≀ ([2], U2) ≺ J5(n+2).

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 54

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Strategy of proof Definition U2 :=

  • ,

,

  • (transformation monoid on [2] := {1, 2})

The proof shows that pvar{Jn | n ∈ N} is closed under the

  • peration

S → S ≀ ([2], U2),

  • r, more precisely:

S ≺ Jn = ⇒ S ≀ ([2], U2) ≺ J5(n+2). The claim then follows from the Krohn–Rhodes Theorem.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition Tm = all transformations of [m] := {1, . . . , m} (acting on the right)

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition Tm = all transformations of [m] := {1, . . . , m} (acting on the right) The elements of Tm can be seen as diagrams like this:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 57

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition Tm = all transformations of [m] := {1, . . . , m} (acting on the right) The elements of Tm can be seen as diagrams like this:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Definition Tm = all transformations of [m] := {1, . . . , m} (acting on the right) The elements of Tm can be seen as diagrams like this: which are composed in the obvious way.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Take any monoid S and T ≤ Tm.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Take any monoid S and T ≤ Tm. The elements of S ≀ ([m], T) are labelled diagrams like these:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Take any monoid S and T ≤ Tm. The elements of S ≀ ([m], T) are labelled diagrams like these: a1 a2 a3 a4 a5 a6 a7 (ai ∈ S)

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Take any monoid S and T ≤ Tm. The elements of S ≀ ([m], T) are labelled diagrams like these: a1 a2 a3 a4 a5 a6 a7 (ai ∈ S) b7 b6 b5 b4 b3 b2 b1 (bi ∈ S)

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Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Take any monoid S and T ≤ Tm. The elements of S ≀ ([m], T) are labelled diagrams like these: a1 a2 a3 a4 a5 a6 a7 (ai ∈ S) b7 b6 b5 b4 b3 b2 b1 (bi ∈ S) a1 a2 a3 a4 a5 a6 a7 (ai ∈ S) b7 b6 b5 b4 b3 b2 b1 (bi ∈ S) = a7b5 a5b4 a6b7 a4b4 a2b4 a1b3 a3b1

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 64

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm?

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 65

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? =

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SLIDE 66

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d

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SLIDE 67

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d d

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SLIDE 68

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d d ?

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 69

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d d ? Solution:

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 70

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d d ? Solution: orient the arcs and label them by elements of a monoid with involution x → xR!

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 71

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

Why don’t we do the same for diagrams in Jm and/or Bm? = c a b d d ? Solution: orient the arcs and label them by elements of a monoid with involution x → xR! = c a b d abcR d

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 72

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

slide-73
SLIDE 73

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm Essential properties: Jn L Jm ֒ → J(n+2)m

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

slide-74
SLIDE 74

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm Essential properties: Tn ≀ ([m], Tm) ֒ → Tnm

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

slide-75
SLIDE 75

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm Essential properties: Jn L Jm ֒ → J(n+2)m

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

slide-76
SLIDE 76

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm Essential properties: Jn L Jm ֒ → J(n+2)m and S ≺ ˜ S = ⇒ S ≀ ([2], U2) ≺ ˜ S L J5.

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 77

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

labelled product S L J of an involutory monoid S and J ≤ Bm Essential properties: Jn L Jm ֒ → J(n+2)m and S ≺ ˜ S = ⇒ S ≀ ([2], U2) ≺ ˜ S L J5. Therefore, S ≺ Jn = ⇒ S ≀ ([2], U2) ≺ Jn L J5 ֒ → J5(n+2).

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 78

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

For the the claim pvar{Bn | n ∈ N} = M

  • ne shows:
  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 79

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

For the the claim pvar{Bn | n ∈ N} = M

  • ne shows:

S ≺ Bn = ⇒ S ≀ ([2], U2) ≺ B5(n+2)

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 80

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

For the the claim pvar{Bn | n ∈ N} = M

  • ne shows:

S ≺ Bn = ⇒ S ≀ ([2], U2) ≺ B5(n+2) and S ≺ Bn = ⇒ S ≀ ([m], Sm) ≺ Bm(n+2).

  • K. Auinger

Pseudovarieties generated by Brauer type monoids

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SLIDE 81

Brauer type monoids Pseudovarieties Main result Context Opposite view Strategy of proof Wreath product and labelled product

For the the claim pvar{Bn | n ∈ N} = M

  • ne shows:

S ≺ Bn = ⇒ S ≀ ([2], U2) ≺ B5(n+2) and S ≺ Bn = ⇒ S ≀ ([m], Sm) ≺ Bm(n+2). Thanks!

  • K. Auinger

Pseudovarieties generated by Brauer type monoids